1. Post clustering inference does not impose additional restrictions on the parameter space. We discuss, in Section REF , clustering methods tailored to model (REF ), where the number of clusters \(K\)  is unknown and is allowed to grow with \(n\) . Using the results of [1]}, these methods yield a partition \(\widehat{G} = G^*\) , with high probability, provided that \(\lambda _{\text{min}}(^*) > c\) , for a small positive quantity \(c\)  made precise in Section REF . A lower bound on the smallest eigenvalue of the covariance matrix is the minimal condition under which inference in any general graphical model can be performed. Therefore, consistent clustering via model (REF ) does not require a further reduction of the parameter space for which the more standard post-regularized inference can be developed. Moreover, as Section  shows, asymptotic inference based on the estimated clusters reduces to asymptotic inference relative to the true clusters, \(G^*\) , without any need for data splitting. This fact is in contrast with the phenomenon encountered in post-model selection inference, for instance in variable selection in linear regression [2]}, [3]}, [4]}. In that case, reducing inference to the consistently selected set of variables can only be justified over a reduced part of the parameter space [5]}.
